# Equipartition theorem for the second moment of the energy

The equipartition theorem provides a convenient way to derive a relation between the hamiltonian of an ideal gas to the temperature of the system. For an extreme relativistic ideal gas, the kinetic energy of a single particle is given by the formula

$$H=c\sqrt{p_x^2+p_y^2+p_z^2}.$$

Then one can write the average

$$\langle H\rangle=\langle p_x\frac{\partial H}{\partial p_x}\rangle+\langle p_y\frac{\partial H}{\partial p_y}\rangle+\langle p_z\frac{\partial H}{\partial p_z}\rangle= 3 k_{\small\text{B}} T$$

where the last equality follows from the equipartition formula.

Is it possible to get a similar relation for the second moment of the energy $$\langle H^2\rangle$$?

Similar to the formula $$\langle p_x \frac{\partial H}{\partial p_x} \rangle = k_BT,$$ there exists the following relation $$\langle p_x \frac{\partial H^2}{\partial p_x} \rangle = 2 k_BT \langle H \rangle + 2(k_B T)^2. \quad (1)$$ For the hamiltonian of an ultrarelativistic particle, we have $$H^2 = \frac12\left(p_x \frac{\partial H^2}{\partial p_x} + p_y \frac{\partial H^2}{\partial p_y} + p_z \frac{\partial H^2}{\partial p_z} \right)$$ Together with $$\langle H \rangle = 3k_BT$$, this gives $$\langle H^2 \rangle = 12(k_BT)^2.$$
• @kaffeeauf, I don't have a reference. I've derived it myself. Do you know how to derive a relation for $<p_x\partial H/\partial p_x>$? The same method gives a relation for $H^2$.
• The formula (1) is general. It is applicable to any reasonable hamiltonian $H$.